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Theorem isfin5 7970
Description: Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin5  |-  ( A  e. FinV  <-> 
( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) )

Proof of Theorem isfin5
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-fin5 7960 . . 3  |- FinV  =  {
x  |  ( x  =  (/)  \/  x  ~<  ( x  +c  x
) ) }
21eleq2i 2380 . 2  |-  ( A  e. FinV  <-> 
A  e.  { x  |  ( x  =  (/)  \/  x  ~<  (
x  +c  x ) ) } )
3 id 19 . . . . 5  |-  ( A  =  (/)  ->  A  =  (/) )
4 0ex 4187 . . . . 5  |-  (/)  e.  _V
53, 4syl6eqel 2404 . . . 4  |-  ( A  =  (/)  ->  A  e. 
_V )
6 relsdom 6913 . . . . 5  |-  Rel  ~<
76brrelexi 4766 . . . 4  |-  ( A 
~<  ( A  +c  A
)  ->  A  e.  _V )
85, 7jaoi 368 . . 3  |-  ( ( A  =  (/)  \/  A  ~<  ( A  +c  A
) )  ->  A  e.  _V )
9 eqeq1 2322 . . . 4  |-  ( x  =  A  ->  (
x  =  (/)  <->  A  =  (/) ) )
10 id 19 . . . . 5  |-  ( x  =  A  ->  x  =  A )
1110, 10oveq12d 5918 . . . . 5  |-  ( x  =  A  ->  (
x  +c  x )  =  ( A  +c  A ) )
1210, 11breq12d 4073 . . . 4  |-  ( x  =  A  ->  (
x  ~<  ( x  +c  x )  <->  A  ~<  ( A  +c  A ) ) )
139, 12orbi12d 690 . . 3  |-  ( x  =  A  ->  (
( x  =  (/)  \/  x  ~<  ( x  +c  x ) )  <->  ( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) ) )
148, 13elab3 2955 . 2  |-  ( A  e.  { x  |  ( x  =  (/)  \/  x  ~<  ( x  +c  x ) ) }  <-> 
( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) )
152, 14bitri 240 1  |-  ( A  e. FinV  <-> 
( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    = wceq 1633    e. wcel 1701   {cab 2302   _Vcvv 2822   (/)c0 3489   class class class wbr 4060  (class class class)co 5900    ~< csdm 6905    +c ccda 7838  FinVcfin5 7953
This theorem is referenced by:  isfin5-2  8062  fin56  8064
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-xp 4732  df-rel 4733  df-iota 5256  df-fv 5300  df-ov 5903  df-dom 6908  df-sdom 6909  df-fin5 7960
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